Toniolo and Linder, Equation (3b), real

Percentage Accurate: 93.7% → 99.7%

Time: 25.1s

Alternatives: 21

Speedup: 1.4×

Specification

Math

\[\begin{array}{l} \\ \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) (sin th)))

C

double code(double kx, double ky, double th) {
	return (sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) * sin(th);
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = (sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) * sin(th)
end function

Java

public static double code(double kx, double ky, double th) {
	return (Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) * Math.sin(th);
}

Python

def code(kx, ky, th):
	return (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) * math.sin(th)

Julia

function code(kx, ky, th)
	return Float64(Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = (sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th);
end

Wolfram

code[kx_, ky_, th_] := N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]

Initial Program: 93.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) (sin th)))

C

double code(double kx, double ky, double th) {
	return (sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) * sin(th);
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = (sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) * sin(th)
end function

Java

public static double code(double kx, double ky, double th) {
	return (Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) * Math.sin(th);
}

Python

def code(kx, ky, th):
	return (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) * math.sin(th)

Julia

function code(kx, ky, th)
	return Float64(Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = (sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th);
end

Wolfram

code[kx_, ky_, th_] := N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (/ (sin th) (/ (hypot (sin kx) (sin ky)) (sin ky))))

C

double code(double kx, double ky, double th) {
	return sin(th) / (hypot(sin(kx), sin(ky)) / sin(ky));
}

Java

public static double code(double kx, double ky, double th) {
	return Math.sin(th) / (Math.hypot(Math.sin(kx), Math.sin(ky)) / Math.sin(ky));
}

Python

def code(kx, ky, th):
	return math.sin(th) / (math.hypot(math.sin(kx), math.sin(ky)) / math.sin(ky))

Julia

function code(kx, ky, th)
	return Float64(sin(th) / Float64(hypot(sin(kx), sin(ky)) / sin(ky)))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = sin(th) / (hypot(sin(kx), sin(ky)) / sin(ky));
end

Wolfram

code[kx_, ky_, th_] := N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.7%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing
3. Step-by-step derivation

   1. clear-num 94.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \cdot \sin th \]

   2. associate-*l/ 94.8%

      \[\leadsto \color{blue}{\frac{1 \cdot \sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

   3. *-un-lft-identity 94.8%

      \[\leadsto \frac{\color{blue}{\sin th}}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}} \]

   4. unpow2 94.8%

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}}{\sin ky}} \]

   5. unpow2 94.8%

      \[\leadsto \frac{\sin th}{\frac{\sqrt{\sin kx \cdot \sin kx + \color{blue}{\sin ky \cdot \sin ky}}}{\sin ky}} \]

   6. hypot-def 99.7%

      \[\leadsto \frac{\sin th}{\frac{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{\sin ky}} \]

4. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}} \]

5. Final simplification 99.7%

   \[\leadsto \frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}} \]
6. Add Preprocessing

Alternative 2: 43.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin th}{\left|\frac{\sin kx}{ky}\right|}\\ \mathbf{if}\;\sin ky \leq -0.001:\\ \;\;\;\;\frac{1}{\sqrt{{\sin th}^{-2}}}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-150}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin ky \leq 10^{-118}:\\ \;\;\;\;\frac{1}{\frac{\sin ky}{\sin th \cdot \sin ky}}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-21}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin th) (fabs (/ (sin kx) ky)))))
   (if (<= (sin ky) -0.001)
     (/ 1.0 (sqrt (pow (sin th) -2.0)))
     (if (<= (sin ky) 2e-150)
       t_1
       (if (<= (sin ky) 1e-118)
         (/ 1.0 (/ (sin ky) (* (sin th) (sin ky))))
         (if (<= (sin ky) 2e-21) t_1 (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(th) / fabs((sin(kx) / ky));
	double tmp;
	if (sin(ky) <= -0.001) {
		tmp = 1.0 / sqrt(pow(sin(th), -2.0));
	} else if (sin(ky) <= 2e-150) {
		tmp = t_1;
	} else if (sin(ky) <= 1e-118) {
		tmp = 1.0 / (sin(ky) / (sin(th) * sin(ky)));
	} else if (sin(ky) <= 2e-21) {
		tmp = t_1;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(th) / abs((sin(kx) / ky))
    if (sin(ky) <= (-0.001d0)) then
        tmp = 1.0d0 / sqrt((sin(th) ** (-2.0d0)))
    else if (sin(ky) <= 2d-150) then
        tmp = t_1
    else if (sin(ky) <= 1d-118) then
        tmp = 1.0d0 / (sin(ky) / (sin(th) * sin(ky)))
    else if (sin(ky) <= 2d-21) then
        tmp = t_1
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(th) / Math.abs((Math.sin(kx) / ky));
	double tmp;
	if (Math.sin(ky) <= -0.001) {
		tmp = 1.0 / Math.sqrt(Math.pow(Math.sin(th), -2.0));
	} else if (Math.sin(ky) <= 2e-150) {
		tmp = t_1;
	} else if (Math.sin(ky) <= 1e-118) {
		tmp = 1.0 / (Math.sin(ky) / (Math.sin(th) * Math.sin(ky)));
	} else if (Math.sin(ky) <= 2e-21) {
		tmp = t_1;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(th) / math.fabs((math.sin(kx) / ky))
	tmp = 0
	if math.sin(ky) <= -0.001:
		tmp = 1.0 / math.sqrt(math.pow(math.sin(th), -2.0))
	elif math.sin(ky) <= 2e-150:
		tmp = t_1
	elif math.sin(ky) <= 1e-118:
		tmp = 1.0 / (math.sin(ky) / (math.sin(th) * math.sin(ky)))
	elif math.sin(ky) <= 2e-21:
		tmp = t_1
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(th) / abs(Float64(sin(kx) / ky)))
	tmp = 0.0
	if (sin(ky) <= -0.001)
		tmp = Float64(1.0 / sqrt((sin(th) ^ -2.0)));
	elseif (sin(ky) <= 2e-150)
		tmp = t_1;
	elseif (sin(ky) <= 1e-118)
		tmp = Float64(1.0 / Float64(sin(ky) / Float64(sin(th) * sin(ky))));
	elseif (sin(ky) <= 2e-21)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(th) / abs((sin(kx) / ky));
	tmp = 0.0;
	if (sin(ky) <= -0.001)
		tmp = 1.0 / sqrt((sin(th) ^ -2.0));
	elseif (sin(ky) <= 2e-150)
		tmp = t_1;
	elseif (sin(ky) <= 1e-118)
		tmp = 1.0 / (sin(ky) / (sin(th) * sin(ky)));
	elseif (sin(ky) <= 2e-21)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[th], $MachinePrecision] / N[Abs[N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sin[ky], $MachinePrecision], -0.001], N[(1.0 / N[Sqrt[N[Power[N[Sin[th], $MachinePrecision], -2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-150], t$95$1, If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-118], N[(1.0 / N[(N[Sin[ky], $MachinePrecision] / N[(N[Sin[th], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-21], t$95$1, N[Sin[th], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 ky) < -1e-3

   1. Initial program 99.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l/ 99.7%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. clear-num 99.4%

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      3. unpow2 99.4%

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}} \]

      4. unpow2 99.4%

         \[\leadsto \frac{1}{\frac{\sqrt{\sin kx \cdot \sin kx + \color{blue}{\sin ky \cdot \sin ky}}}{\sin ky \cdot \sin th}} \]

      5. hypot-def 99.4%

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{\sin ky \cdot \sin th}} \]

   4. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot \sin th}}} \]

   5. Taylor expanded in kx around 0 3.1%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\sin th}}} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 1.8%

         \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{1}{\sin th}} \cdot \sqrt{\frac{1}{\sin th}}}} \]

      2. sqrt-unprod 22.4%

         \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{1}{\sin th} \cdot \frac{1}{\sin th}}}} \]

      3. inv-pow 22.4%

         \[\leadsto \frac{1}{\sqrt{\color{blue}{{\sin th}^{-1}} \cdot \frac{1}{\sin th}}} \]

      4. inv-pow 22.4%

         \[\leadsto \frac{1}{\sqrt{{\sin th}^{-1} \cdot \color{blue}{{\sin th}^{-1}}}} \]

      5. pow-prod-up 22.4%

         \[\leadsto \frac{1}{\sqrt{\color{blue}{{\sin th}^{\left(-1 + -1\right)}}}} \]

      6. metadata-eval 22.4%

         \[\leadsto \frac{1}{\sqrt{{\sin th}^{\color{blue}{-2}}}} \]

   7. Applied egg-rr 22.4%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{{\sin th}^{-2}}}} \]

   if -1e-3 < (sin.f64 ky) < 2.00000000000000001e-150 or 9.99999999999999985e-119 < (sin.f64 ky) < 1.99999999999999982e-21

   1. Initial program 88.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 88.3%

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \cdot \sin th \]

      2. associate-*l/ 88.3%

         \[\leadsto \color{blue}{\frac{1 \cdot \sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      3. *-un-lft-identity 88.3%

         \[\leadsto \frac{\color{blue}{\sin th}}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}} \]

      4. unpow2 88.3%

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}}{\sin ky}} \]

      5. unpow2 88.3%

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\sin kx \cdot \sin kx + \color{blue}{\sin ky \cdot \sin ky}}}{\sin ky}} \]

      6. hypot-def 99.7%

         \[\leadsto \frac{\sin th}{\frac{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{\sin ky}} \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}} \]

   5. Taylor expanded in ky around 0 51.2%

      \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sin kx}{ky}}} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 25.5%

         \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{\frac{\sin kx}{ky}} \cdot \sqrt{\frac{\sin kx}{ky}}}} \]

      2. sqrt-unprod 46.1%

         \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{\frac{\sin kx}{ky} \cdot \frac{\sin kx}{ky}}}} \]

      3. pow2 46.1%

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\left(\frac{\sin kx}{ky}\right)}^{2}}}} \]

   7. Applied egg-rr 46.1%

      \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{{\left(\frac{\sin kx}{ky}\right)}^{2}}}} \]
   8. Step-by-step derivation

      1. unpow2 46.1%

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\frac{\sin kx}{ky} \cdot \frac{\sin kx}{ky}}}} \]

      2. rem-sqrt-square 55.1%

         \[\leadsto \frac{\sin th}{\color{blue}{\left|\frac{\sin kx}{ky}\right|}} \]

   9. Simplified 55.1%

      \[\leadsto \frac{\sin th}{\color{blue}{\left|\frac{\sin kx}{ky}\right|}} \]

   if 2.00000000000000001e-150 < (sin.f64 ky) < 9.99999999999999985e-119

   1. Initial program 100.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l/ 80.5%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. clear-num 80.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      3. unpow2 80.7%

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}} \]

      4. unpow2 80.7%

         \[\leadsto \frac{1}{\frac{\sqrt{\sin kx \cdot \sin kx + \color{blue}{\sin ky \cdot \sin ky}}}{\sin ky \cdot \sin th}} \]

      5. hypot-def 80.7%

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{\sin ky \cdot \sin th}} \]

   4. Applied egg-rr 80.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot \sin th}}} \]

   5. Taylor expanded in kx around 0 80.7%

      \[\leadsto \frac{1}{\frac{\color{blue}{\sin ky}}{\sin ky \cdot \sin th}} \]

   if 1.99999999999999982e-21 < (sin.f64 ky)

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0 63.9%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 4 regimes into one program.

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.001:\\ \;\;\;\;\frac{1}{\sqrt{{\sin th}^{-2}}}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-150}:\\ \;\;\;\;\frac{\sin th}{\left|\frac{\sin kx}{ky}\right|}\\ \mathbf{elif}\;\sin ky \leq 10^{-118}:\\ \;\;\;\;\frac{1}{\frac{\sin ky}{\sin th \cdot \sin ky}}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-21}:\\ \;\;\;\;\frac{\sin th}{\left|\frac{\sin kx}{ky}\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 41.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.38:\\ \;\;\;\;\left|ky \cdot \frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 10^{-166}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-60}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.38)
   (fabs (* ky (/ (sin th) (sin kx))))
   (if (<= (sin kx) 1e-166)
     (sin th)
     (if (<= (sin kx) 2e-60)
       (fabs (sin th))
       (* (sin th) (/ (sin ky) (sin kx)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.38) {
		tmp = fabs((ky * (sin(th) / sin(kx))));
	} else if (sin(kx) <= 1e-166) {
		tmp = sin(th);
	} else if (sin(kx) <= 2e-60) {
		tmp = fabs(sin(th));
	} else {
		tmp = sin(th) * (sin(ky) / sin(kx));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(kx) <= (-0.38d0)) then
        tmp = abs((ky * (sin(th) / sin(kx))))
    else if (sin(kx) <= 1d-166) then
        tmp = sin(th)
    else if (sin(kx) <= 2d-60) then
        tmp = abs(sin(th))
    else
        tmp = sin(th) * (sin(ky) / sin(kx))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.38) {
		tmp = Math.abs((ky * (Math.sin(th) / Math.sin(kx))));
	} else if (Math.sin(kx) <= 1e-166) {
		tmp = Math.sin(th);
	} else if (Math.sin(kx) <= 2e-60) {
		tmp = Math.abs(Math.sin(th));
	} else {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.38:
		tmp = math.fabs((ky * (math.sin(th) / math.sin(kx))))
	elif math.sin(kx) <= 1e-166:
		tmp = math.sin(th)
	elif math.sin(kx) <= 2e-60:
		tmp = math.fabs(math.sin(th))
	else:
		tmp = math.sin(th) * (math.sin(ky) / math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.38)
		tmp = abs(Float64(ky * Float64(sin(th) / sin(kx))));
	elseif (sin(kx) <= 1e-166)
		tmp = sin(th);
	elseif (sin(kx) <= 2e-60)
		tmp = abs(sin(th));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) / sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.38)
		tmp = abs((ky * (sin(th) / sin(kx))));
	elseif (sin(kx) <= 1e-166)
		tmp = sin(th);
	elseif (sin(kx) <= 2e-60)
		tmp = abs(sin(th));
	else
		tmp = sin(th) * (sin(ky) / sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.38], N[Abs[N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 1e-166], N[Sin[th], $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 2e-60], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 kx) < -0.38

   1. Initial program 99.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 99.4%

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \cdot \sin th \]

      2. associate-*l/ 99.5%

         \[\leadsto \color{blue}{\frac{1 \cdot \sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      3. *-un-lft-identity 99.5%

         \[\leadsto \frac{\color{blue}{\sin th}}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}} \]

      4. unpow2 99.5%

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}}{\sin ky}} \]

      5. unpow2 99.5%

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\sin kx \cdot \sin kx + \color{blue}{\sin ky \cdot \sin ky}}}{\sin ky}} \]

      6. hypot-def 99.5%

         \[\leadsto \frac{\sin th}{\frac{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{\sin ky}} \]

   4. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}} \]

   5. Taylor expanded in ky around 0 15.0%

      \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sin kx}{ky}}} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 13.2%

         \[\leadsto \color{blue}{\sqrt{\frac{\sin th}{\frac{\sin kx}{ky}}} \cdot \sqrt{\frac{\sin th}{\frac{\sin kx}{ky}}}} \]

      2. sqrt-unprod 22.7%

         \[\leadsto \color{blue}{\sqrt{\frac{\sin th}{\frac{\sin kx}{ky}} \cdot \frac{\sin th}{\frac{\sin kx}{ky}}}} \]

      3. pow2 22.7%

         \[\leadsto \sqrt{\color{blue}{{\left(\frac{\sin th}{\frac{\sin kx}{ky}}\right)}^{2}}} \]

      4. associate-/r/ 22.7%

         \[\leadsto \sqrt{{\color{blue}{\left(\frac{\sin th}{\sin kx} \cdot ky\right)}}^{2}} \]

      5. *-commutative 22.7%

         \[\leadsto \sqrt{{\color{blue}{\left(ky \cdot \frac{\sin th}{\sin kx}\right)}}^{2}} \]

   7. Applied egg-rr 22.7%

      \[\leadsto \color{blue}{\sqrt{{\left(ky \cdot \frac{\sin th}{\sin kx}\right)}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 22.7%

         \[\leadsto \sqrt{\color{blue}{\left(ky \cdot \frac{\sin th}{\sin kx}\right) \cdot \left(ky \cdot \frac{\sin th}{\sin kx}\right)}} \]

      2. rem-sqrt-square 33.3%

         \[\leadsto \color{blue}{\left|ky \cdot \frac{\sin th}{\sin kx}\right|} \]

   9. Simplified 33.3%

      \[\leadsto \color{blue}{\left|ky \cdot \frac{\sin th}{\sin kx}\right|} \]

   if -0.38 < (sin.f64 kx) < 1.00000000000000004e-166

   1. Initial program 88.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0 41.7%

      \[\leadsto \color{blue}{\sin th} \]

   if 1.00000000000000004e-166 < (sin.f64 kx) < 1.9999999999999999e-60

   1. Initial program 99.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l/ 95.7%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. clear-num 95.4%

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      3. unpow2 95.4%

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}} \]

      4. unpow2 95.4%

         \[\leadsto \frac{1}{\frac{\sqrt{\sin kx \cdot \sin kx + \color{blue}{\sin ky \cdot \sin ky}}}{\sin ky \cdot \sin th}} \]

      5. hypot-def 95.4%

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{\sin ky \cdot \sin th}} \]

   4. Applied egg-rr 95.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot \sin th}}} \]

   5. Taylor expanded in kx around 0 34.4%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\sin th}}} \]
   6. Step-by-step derivation

      1. remove-double-div 34.4%

         \[\leadsto \color{blue}{\sin th} \]

      2. add-sqr-sqrt 28.9%

         \[\leadsto \color{blue}{\sqrt{\sin th} \cdot \sqrt{\sin th}} \]

      3. sqrt-unprod 22.3%

         \[\leadsto \color{blue}{\sqrt{\sin th \cdot \sin th}} \]

      4. pow2 22.3%

         \[\leadsto \sqrt{\color{blue}{{\sin th}^{2}}} \]

   7. Applied egg-rr 22.3%

      \[\leadsto \color{blue}{\sqrt{{\sin th}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 22.3%

         \[\leadsto \sqrt{\color{blue}{\sin th \cdot \sin th}} \]

      2. rem-sqrt-square 43.2%

         \[\leadsto \color{blue}{\left|\sin th\right|} \]

   9. Simplified 43.2%

      \[\leadsto \color{blue}{\left|\sin th\right|} \]

   if 1.9999999999999999e-60 < (sin.f64 kx)

   1. Initial program 99.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0 60.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
3. Recombined 4 regimes into one program.

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.38:\\ \;\;\;\;\left|ky \cdot \frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 10^{-166}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-60}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.001:\\ \;\;\;\;\frac{1}{\sqrt{{\sin th}^{-2}}}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-21}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin th \cdot ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.001)
   (/ 1.0 (sqrt (pow (sin th) -2.0)))
   (if (<= (sin ky) 2e-21)
     (/ 1.0 (/ (hypot (sin kx) (sin ky)) (* (sin th) ky)))
     (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.001) {
		tmp = 1.0 / sqrt(pow(sin(th), -2.0));
	} else if (sin(ky) <= 2e-21) {
		tmp = 1.0 / (hypot(sin(kx), sin(ky)) / (sin(th) * ky));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.001) {
		tmp = 1.0 / Math.sqrt(Math.pow(Math.sin(th), -2.0));
	} else if (Math.sin(ky) <= 2e-21) {
		tmp = 1.0 / (Math.hypot(Math.sin(kx), Math.sin(ky)) / (Math.sin(th) * ky));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.001:
		tmp = 1.0 / math.sqrt(math.pow(math.sin(th), -2.0))
	elif math.sin(ky) <= 2e-21:
		tmp = 1.0 / (math.hypot(math.sin(kx), math.sin(ky)) / (math.sin(th) * ky))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.001)
		tmp = Float64(1.0 / sqrt((sin(th) ^ -2.0)));
	elseif (sin(ky) <= 2e-21)
		tmp = Float64(1.0 / Float64(hypot(sin(kx), sin(ky)) / Float64(sin(th) * ky)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.001)
		tmp = 1.0 / sqrt((sin(th) ^ -2.0));
	elseif (sin(ky) <= 2e-21)
		tmp = 1.0 / (hypot(sin(kx), sin(ky)) / (sin(th) * ky));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.001], N[(1.0 / N[Sqrt[N[Power[N[Sin[th], $MachinePrecision], -2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-21], N[(1.0 / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -1e-3

   1. Initial program 99.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l/ 99.7%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. clear-num 99.4%

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      3. unpow2 99.4%

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}} \]

      4. unpow2 99.4%

         \[\leadsto \frac{1}{\frac{\sqrt{\sin kx \cdot \sin kx + \color{blue}{\sin ky \cdot \sin ky}}}{\sin ky \cdot \sin th}} \]

      5. hypot-def 99.4%

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{\sin ky \cdot \sin th}} \]

   4. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot \sin th}}} \]

   5. Taylor expanded in kx around 0 3.1%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\sin th}}} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 1.8%

         \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{1}{\sin th}} \cdot \sqrt{\frac{1}{\sin th}}}} \]

      2. sqrt-unprod 22.4%

         \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{1}{\sin th} \cdot \frac{1}{\sin th}}}} \]

      3. inv-pow 22.4%

         \[\leadsto \frac{1}{\sqrt{\color{blue}{{\sin th}^{-1}} \cdot \frac{1}{\sin th}}} \]

      4. inv-pow 22.4%

         \[\leadsto \frac{1}{\sqrt{{\sin th}^{-1} \cdot \color{blue}{{\sin th}^{-1}}}} \]

      5. pow-prod-up 22.4%

         \[\leadsto \frac{1}{\sqrt{\color{blue}{{\sin th}^{\left(-1 + -1\right)}}}} \]

      6. metadata-eval 22.4%

         \[\leadsto \frac{1}{\sqrt{{\sin th}^{\color{blue}{-2}}}} \]

   7. Applied egg-rr 22.4%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{{\sin th}^{-2}}}} \]

   if -1e-3 < (sin.f64 ky) < 1.99999999999999982e-21

   1. Initial program 88.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l/ 86.5%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. clear-num 85.8%

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      3. unpow2 85.8%

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}} \]

      4. unpow2 85.8%

         \[\leadsto \frac{1}{\frac{\sqrt{\sin kx \cdot \sin kx + \color{blue}{\sin ky \cdot \sin ky}}}{\sin ky \cdot \sin th}} \]

      5. hypot-def 94.3%

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{\sin ky \cdot \sin th}} \]

   4. Applied egg-rr 94.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot \sin th}}} \]

   5. Taylor expanded in ky around 0 94.3%

      \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\color{blue}{ky \cdot \sin th}}} \]

   if 1.99999999999999982e-21 < (sin.f64 ky)

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0 63.9%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.001:\\ \;\;\;\;\frac{1}{\sqrt{{\sin th}^{-2}}}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-21}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin th \cdot ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.001:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{th \cdot \sin ky}}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-21}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin th \cdot ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.001)
   (/ 1.0 (/ (hypot (sin ky) (sin kx)) (* th (sin ky))))
   (if (<= (sin ky) 2e-21)
     (/ 1.0 (/ (hypot (sin kx) (sin ky)) (* (sin th) ky)))
     (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.001) {
		tmp = 1.0 / (hypot(sin(ky), sin(kx)) / (th * sin(ky)));
	} else if (sin(ky) <= 2e-21) {
		tmp = 1.0 / (hypot(sin(kx), sin(ky)) / (sin(th) * ky));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.001) {
		tmp = 1.0 / (Math.hypot(Math.sin(ky), Math.sin(kx)) / (th * Math.sin(ky)));
	} else if (Math.sin(ky) <= 2e-21) {
		tmp = 1.0 / (Math.hypot(Math.sin(kx), Math.sin(ky)) / (Math.sin(th) * ky));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.001:
		tmp = 1.0 / (math.hypot(math.sin(ky), math.sin(kx)) / (th * math.sin(ky)))
	elif math.sin(ky) <= 2e-21:
		tmp = 1.0 / (math.hypot(math.sin(kx), math.sin(ky)) / (math.sin(th) * ky))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.001)
		tmp = Float64(1.0 / Float64(hypot(sin(ky), sin(kx)) / Float64(th * sin(ky))));
	elseif (sin(ky) <= 2e-21)
		tmp = Float64(1.0 / Float64(hypot(sin(kx), sin(ky)) / Float64(sin(th) * ky)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.001)
		tmp = 1.0 / (hypot(sin(ky), sin(kx)) / (th * sin(ky)));
	elseif (sin(ky) <= 2e-21)
		tmp = 1.0 / (hypot(sin(kx), sin(ky)) / (sin(th) * ky));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.001], N[(1.0 / N[(N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision] / N[(th * N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-21], N[(1.0 / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -1e-3

   1. Initial program 99.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l/ 99.7%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. clear-num 99.4%

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      3. unpow2 99.4%

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}} \]

      4. unpow2 99.4%

         \[\leadsto \frac{1}{\frac{\sqrt{\sin kx \cdot \sin kx + \color{blue}{\sin ky \cdot \sin ky}}}{\sin ky \cdot \sin th}} \]

      5. hypot-def 99.4%

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{\sin ky \cdot \sin th}} \]

   4. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot \sin th}}} \]

   5. Taylor expanded in th around 0 62.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{th \cdot \sin ky} \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]
   6. Step-by-step derivation

      1. associate-*l/ 62.4%

         \[\leadsto \frac{1}{\color{blue}{\frac{1 \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{th \cdot \sin ky}}} \]

      2. unpow2 62.4%

         \[\leadsto \frac{1}{\frac{1 \cdot \sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}}{th \cdot \sin ky}} \]

      3. unpow2 62.4%

         \[\leadsto \frac{1}{\frac{1 \cdot \sqrt{\sin kx \cdot \sin kx + \color{blue}{\sin ky \cdot \sin ky}}}{th \cdot \sin ky}} \]

      4. hypot-def 62.4%

         \[\leadsto \frac{1}{\frac{1 \cdot \color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{th \cdot \sin ky}} \]

      5. *-lft-identity 62.4%

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{th \cdot \sin ky}} \]

      6. hypot-def 62.4%

         \[\leadsto \frac{1}{\frac{\color{blue}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}}{th \cdot \sin ky}} \]

      7. unpow2 62.4%

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{{\sin kx}^{2}} + \sin ky \cdot \sin ky}}{th \cdot \sin ky}} \]

      8. unpow2 62.4%

         \[\leadsto \frac{1}{\frac{\sqrt{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}}{th \cdot \sin ky}} \]

      9. +-commutative 62.4%

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}}{th \cdot \sin ky}} \]

      10. unpow2 62.4%

          \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}}{th \cdot \sin ky}} \]

      11. unpow2 62.4%

          \[\leadsto \frac{1}{\frac{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}}{th \cdot \sin ky}} \]

      12. hypot-def 62.4%

          \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}{th \cdot \sin ky}} \]

   7. Simplified 62.4%

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{th \cdot \sin ky}}} \]

   if -1e-3 < (sin.f64 ky) < 1.99999999999999982e-21

   1. Initial program 88.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l/ 86.5%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. clear-num 85.8%

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      3. unpow2 85.8%

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}} \]

      4. unpow2 85.8%

         \[\leadsto \frac{1}{\frac{\sqrt{\sin kx \cdot \sin kx + \color{blue}{\sin ky \cdot \sin ky}}}{\sin ky \cdot \sin th}} \]

      5. hypot-def 94.3%

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{\sin ky \cdot \sin th}} \]

   4. Applied egg-rr 94.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot \sin th}}} \]

   5. Taylor expanded in ky around 0 94.3%

      \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\color{blue}{ky \cdot \sin th}}} \]

   if 1.99999999999999982e-21 < (sin.f64 ky)

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0 63.9%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.001:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{th \cdot \sin ky}}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-21}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin th \cdot ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.001:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{th \cdot \sin ky}}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-21}:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \frac{1}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.001)
   (/ 1.0 (/ (hypot (sin ky) (sin kx)) (* th (sin ky))))
   (if (<= (sin ky) 2e-21)
     (/ (sin th) (* (hypot (sin kx) (sin ky)) (/ 1.0 ky)))
     (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.001) {
		tmp = 1.0 / (hypot(sin(ky), sin(kx)) / (th * sin(ky)));
	} else if (sin(ky) <= 2e-21) {
		tmp = sin(th) / (hypot(sin(kx), sin(ky)) * (1.0 / ky));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.001) {
		tmp = 1.0 / (Math.hypot(Math.sin(ky), Math.sin(kx)) / (th * Math.sin(ky)));
	} else if (Math.sin(ky) <= 2e-21) {
		tmp = Math.sin(th) / (Math.hypot(Math.sin(kx), Math.sin(ky)) * (1.0 / ky));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.001:
		tmp = 1.0 / (math.hypot(math.sin(ky), math.sin(kx)) / (th * math.sin(ky)))
	elif math.sin(ky) <= 2e-21:
		tmp = math.sin(th) / (math.hypot(math.sin(kx), math.sin(ky)) * (1.0 / ky))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.001)
		tmp = Float64(1.0 / Float64(hypot(sin(ky), sin(kx)) / Float64(th * sin(ky))));
	elseif (sin(ky) <= 2e-21)
		tmp = Float64(sin(th) / Float64(hypot(sin(kx), sin(ky)) * Float64(1.0 / ky)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.001)
		tmp = 1.0 / (hypot(sin(ky), sin(kx)) / (th * sin(ky)));
	elseif (sin(ky) <= 2e-21)
		tmp = sin(th) / (hypot(sin(kx), sin(ky)) * (1.0 / ky));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.001], N[(1.0 / N[(N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision] / N[(th * N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-21], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] * N[(1.0 / ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -1e-3

   1. Initial program 99.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l/ 99.7%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. clear-num 99.4%

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      3. unpow2 99.4%

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}} \]

      4. unpow2 99.4%

         \[\leadsto \frac{1}{\frac{\sqrt{\sin kx \cdot \sin kx + \color{blue}{\sin ky \cdot \sin ky}}}{\sin ky \cdot \sin th}} \]

      5. hypot-def 99.4%

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{\sin ky \cdot \sin th}} \]

   4. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot \sin th}}} \]

   5. Taylor expanded in th around 0 62.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{th \cdot \sin ky} \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]
   6. Step-by-step derivation

      1. associate-*l/ 62.4%

         \[\leadsto \frac{1}{\color{blue}{\frac{1 \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{th \cdot \sin ky}}} \]

      2. unpow2 62.4%

         \[\leadsto \frac{1}{\frac{1 \cdot \sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}}{th \cdot \sin ky}} \]

      3. unpow2 62.4%

         \[\leadsto \frac{1}{\frac{1 \cdot \sqrt{\sin kx \cdot \sin kx + \color{blue}{\sin ky \cdot \sin ky}}}{th \cdot \sin ky}} \]

      4. hypot-def 62.4%

         \[\leadsto \frac{1}{\frac{1 \cdot \color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{th \cdot \sin ky}} \]

      5. *-lft-identity 62.4%

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{th \cdot \sin ky}} \]

      6. hypot-def 62.4%

         \[\leadsto \frac{1}{\frac{\color{blue}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}}{th \cdot \sin ky}} \]

      7. unpow2 62.4%

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{{\sin kx}^{2}} + \sin ky \cdot \sin ky}}{th \cdot \sin ky}} \]

      8. unpow2 62.4%

         \[\leadsto \frac{1}{\frac{\sqrt{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}}{th \cdot \sin ky}} \]

      9. +-commutative 62.4%

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}}{th \cdot \sin ky}} \]

      10. unpow2 62.4%

          \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}}{th \cdot \sin ky}} \]

      11. unpow2 62.4%

          \[\leadsto \frac{1}{\frac{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}}{th \cdot \sin ky}} \]

      12. hypot-def 62.4%

          \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}{th \cdot \sin ky}} \]

   7. Simplified 62.4%

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{th \cdot \sin ky}}} \]

   if -1e-3 < (sin.f64 ky) < 1.99999999999999982e-21

   1. Initial program 88.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 88.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \cdot \sin th \]

      2. associate-*l/ 88.9%

         \[\leadsto \color{blue}{\frac{1 \cdot \sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      3. *-un-lft-identity 88.9%

         \[\leadsto \frac{\color{blue}{\sin th}}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}} \]

      4. unpow2 88.9%

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}}{\sin ky}} \]

      5. unpow2 88.9%

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\sin kx \cdot \sin kx + \color{blue}{\sin ky \cdot \sin ky}}}{\sin ky}} \]

      6. hypot-def 99.7%

         \[\leadsto \frac{\sin th}{\frac{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{\sin ky}} \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}} \]
   5. Step-by-step derivation

      1. div-inv 99.6%

         \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \frac{1}{\sin ky}}} \]

   6. Applied egg-rr 99.6%

      \[\leadsto \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \frac{1}{\sin ky}}} \]

   7. Taylor expanded in ky around 0 99.6%

      \[\leadsto \frac{\sin th}{\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \color{blue}{\frac{1}{ky}}} \]

   if 1.99999999999999982e-21 < (sin.f64 ky)

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0 63.9%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.001:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{th \cdot \sin ky}}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-21}:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \frac{1}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 44.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.001:\\ \;\;\;\;\frac{1}{\sqrt{{\sin th}^{-2}}}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-21}:\\ \;\;\;\;\frac{\sin th}{\left|\frac{\sin kx}{ky}\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.001)
   (/ 1.0 (sqrt (pow (sin th) -2.0)))
   (if (<= (sin ky) 2e-21) (/ (sin th) (fabs (/ (sin kx) ky))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.001) {
		tmp = 1.0 / sqrt(pow(sin(th), -2.0));
	} else if (sin(ky) <= 2e-21) {
		tmp = sin(th) / fabs((sin(kx) / ky));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-0.001d0)) then
        tmp = 1.0d0 / sqrt((sin(th) ** (-2.0d0)))
    else if (sin(ky) <= 2d-21) then
        tmp = sin(th) / abs((sin(kx) / ky))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.001) {
		tmp = 1.0 / Math.sqrt(Math.pow(Math.sin(th), -2.0));
	} else if (Math.sin(ky) <= 2e-21) {
		tmp = Math.sin(th) / Math.abs((Math.sin(kx) / ky));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.001:
		tmp = 1.0 / math.sqrt(math.pow(math.sin(th), -2.0))
	elif math.sin(ky) <= 2e-21:
		tmp = math.sin(th) / math.fabs((math.sin(kx) / ky))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.001)
		tmp = Float64(1.0 / sqrt((sin(th) ^ -2.0)));
	elseif (sin(ky) <= 2e-21)
		tmp = Float64(sin(th) / abs(Float64(sin(kx) / ky)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.001)
		tmp = 1.0 / sqrt((sin(th) ^ -2.0));
	elseif (sin(ky) <= 2e-21)
		tmp = sin(th) / abs((sin(kx) / ky));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.001], N[(1.0 / N[Sqrt[N[Power[N[Sin[th], $MachinePrecision], -2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 2e-21], N[(N[Sin[th], $MachinePrecision] / N[Abs[N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -1e-3

   1. Initial program 99.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l/ 99.7%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. clear-num 99.4%

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      3. unpow2 99.4%

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}} \]

      4. unpow2 99.4%

         \[\leadsto \frac{1}{\frac{\sqrt{\sin kx \cdot \sin kx + \color{blue}{\sin ky \cdot \sin ky}}}{\sin ky \cdot \sin th}} \]

      5. hypot-def 99.4%

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{\sin ky \cdot \sin th}} \]

   4. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot \sin th}}} \]

   5. Taylor expanded in kx around 0 3.1%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\sin th}}} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 1.8%

         \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{1}{\sin th}} \cdot \sqrt{\frac{1}{\sin th}}}} \]

      2. sqrt-unprod 22.4%

         \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{1}{\sin th} \cdot \frac{1}{\sin th}}}} \]

      3. inv-pow 22.4%

         \[\leadsto \frac{1}{\sqrt{\color{blue}{{\sin th}^{-1}} \cdot \frac{1}{\sin th}}} \]

      4. inv-pow 22.4%

         \[\leadsto \frac{1}{\sqrt{{\sin th}^{-1} \cdot \color{blue}{{\sin th}^{-1}}}} \]

      5. pow-prod-up 22.4%

         \[\leadsto \frac{1}{\sqrt{\color{blue}{{\sin th}^{\left(-1 + -1\right)}}}} \]

      6. metadata-eval 22.4%

         \[\leadsto \frac{1}{\sqrt{{\sin th}^{\color{blue}{-2}}}} \]

   7. Applied egg-rr 22.4%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{{\sin th}^{-2}}}} \]

   if -1e-3 < (sin.f64 ky) < 1.99999999999999982e-21

   1. Initial program 88.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 88.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \cdot \sin th \]

      2. associate-*l/ 88.9%

         \[\leadsto \color{blue}{\frac{1 \cdot \sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      3. *-un-lft-identity 88.9%

         \[\leadsto \frac{\color{blue}{\sin th}}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}} \]

      4. unpow2 88.9%

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}}{\sin ky}} \]

      5. unpow2 88.9%

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\sin kx \cdot \sin kx + \color{blue}{\sin ky \cdot \sin ky}}}{\sin ky}} \]

      6. hypot-def 99.7%

         \[\leadsto \frac{\sin th}{\frac{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{\sin ky}} \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}} \]

   5. Taylor expanded in ky around 0 49.6%

      \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sin kx}{ky}}} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 25.3%

         \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{\frac{\sin kx}{ky}} \cdot \sqrt{\frac{\sin kx}{ky}}}} \]

      2. sqrt-unprod 44.9%

         \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{\frac{\sin kx}{ky} \cdot \frac{\sin kx}{ky}}}} \]

      3. pow2 44.9%

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\left(\frac{\sin kx}{ky}\right)}^{2}}}} \]

   7. Applied egg-rr 44.9%

      \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{{\left(\frac{\sin kx}{ky}\right)}^{2}}}} \]
   8. Step-by-step derivation

      1. unpow2 44.9%

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\frac{\sin kx}{ky} \cdot \frac{\sin kx}{ky}}}} \]

      2. rem-sqrt-square 53.5%

         \[\leadsto \frac{\sin th}{\color{blue}{\left|\frac{\sin kx}{ky}\right|}} \]

   9. Simplified 53.5%

      \[\leadsto \frac{\sin th}{\color{blue}{\left|\frac{\sin kx}{ky}\right|}} \]

   if 1.99999999999999982e-21 < (sin.f64 ky)

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0 63.9%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.001:\\ \;\;\;\;\frac{1}{\sqrt{{\sin th}^{-2}}}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-21}:\\ \;\;\;\;\frac{\sin th}{\left|\frac{\sin kx}{ky}\right|}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (* (sin ky) (/ (sin th) (hypot (sin ky) (sin kx)))))

C

double code(double kx, double ky, double th) {
	return sin(ky) * (sin(th) / hypot(sin(ky), sin(kx)));
}

Java

public static double code(double kx, double ky, double th) {
	return Math.sin(ky) * (Math.sin(th) / Math.hypot(Math.sin(ky), Math.sin(kx)));
}

Python

def code(kx, ky, th):
	return math.sin(ky) * (math.sin(th) / math.hypot(math.sin(ky), math.sin(kx)))

Julia

function code(kx, ky, th)
	return Float64(sin(ky) * Float64(sin(th) / hypot(sin(ky), sin(kx))))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = sin(ky) * (sin(th) / hypot(sin(ky), sin(kx)));
end

Wolfram

code[kx_, ky_, th_] := N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.7%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing
3. Step-by-step derivation

   1. expm1-log1p-u 94.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th\right)\right)} \]

   2. expm1-udef 42.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th\right)} - 1} \]

4. Applied egg-rr 43.7%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin th\right)} - 1} \]
5. Step-by-step derivation

   1. expm1-def 99.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin th\right)\right)} \]

   2. expm1-log1p 99.7%

      \[\leadsto \color{blue}{\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin th} \]

   3. *-commutative 99.7%

      \[\leadsto \color{blue}{\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}} \]

   4. associate-*r/ 97.5%

      \[\leadsto \color{blue}{\frac{\sin th \cdot \sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}} \]

   5. associate-*l/ 99.6%

      \[\leadsto \color{blue}{\frac{\sin th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky} \]

   6. *-commutative 99.6%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}} \]

   7. hypot-def 94.6%

      \[\leadsto \sin ky \cdot \frac{\sin th}{\color{blue}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}} \]

   8. unpow2 94.6%

      \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin kx}^{2}} + \sin ky \cdot \sin ky}} \]

   9. unpow2 94.6%

      \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}} \]

   10. +-commutative 94.6%

       \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \]

   11. unpow2 94.6%

       \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \]

   12. unpow2 94.6%

       \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \]

   13. hypot-def 99.6%

       \[\leadsto \sin ky \cdot \frac{\sin th}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

6. Simplified 99.6%

   \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]

7. Final simplification 99.6%

   \[\leadsto \sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]
8. Add Preprocessing

Alternative 9: 99.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (* (sin th) (/ (sin ky) (hypot (sin ky) (sin kx)))))

C

double code(double kx, double ky, double th) {
	return sin(th) * (sin(ky) / hypot(sin(ky), sin(kx)));
}

Java

public static double code(double kx, double ky, double th) {
	return Math.sin(th) * (Math.sin(ky) / Math.hypot(Math.sin(ky), Math.sin(kx)));
}

Python

def code(kx, ky, th):
	return math.sin(th) * (math.sin(ky) / math.hypot(math.sin(ky), math.sin(kx)))

Julia

function code(kx, ky, th)
	return Float64(sin(th) * Float64(sin(ky) / hypot(sin(ky), sin(kx))))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = sin(th) * (sin(ky) / hypot(sin(ky), sin(kx)));
end

Wolfram

code[kx_, ky_, th_] := N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.7%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Step-by-step derivation

   1. remove-double-neg 94.7%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \color{blue}{\left(-\left(-\sin th\right)\right)} \]

   2. sin-neg 94.7%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \left(-\color{blue}{\sin \left(-th\right)}\right) \]

   3. neg-mul-1 94.7%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \color{blue}{\left(-1 \cdot \sin \left(-th\right)\right)} \]

   4. *-commutative 94.7%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \color{blue}{\left(\sin \left(-th\right) \cdot -1\right)} \]

   5. associate-*l* 94.7%

      \[\leadsto \color{blue}{\left(\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin \left(-th\right)\right) \cdot -1} \]

   6. associate-*l/ 93.6%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin \left(-th\right)}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot -1 \]

   7. associate-/r/ 93.6%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin \left(-th\right)}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{-1}}} \]

   8. associate-*l/ 94.7%

      \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{-1}} \cdot \sin \left(-th\right)} \]

   9. associate-/r/ 94.7%

      \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{-1}}{\sin \left(-th\right)}}} \]

   10. sin-neg 94.7%

       \[\leadsto \frac{\sin ky}{\frac{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{-1}}{\color{blue}{-\sin th}}} \]

   11. neg-mul-1 94.7%

       \[\leadsto \frac{\sin ky}{\frac{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{-1}}{\color{blue}{-1 \cdot \sin th}}} \]

   12. associate-/r* 94.7%

       \[\leadsto \frac{\sin ky}{\color{blue}{\frac{\frac{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{-1}}{-1}}{\sin th}}} \]

   13. associate-/r/ 94.7%

       \[\leadsto \color{blue}{\frac{\sin ky}{\frac{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{-1}}{-1}} \cdot \sin th} \]

3. Simplified 99.7%

   \[\leadsto \color{blue}{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th} \]
4. Add Preprocessing

5. Final simplification 99.7%

   \[\leadsto \sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]
6. Add Preprocessing

Alternative 10: 46.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.178:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-91}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.178)
   (fabs (sin th))
   (if (<= (sin ky) 5e-91) (* (sin th) (/ ky (sin kx))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.178) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 5e-91) {
		tmp = sin(th) * (ky / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-0.178d0)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 5d-91) then
        tmp = sin(th) * (ky / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.178) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 5e-91) {
		tmp = Math.sin(th) * (ky / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.178:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 5e-91:
		tmp = math.sin(th) * (ky / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.178)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 5e-91)
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.178)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 5e-91)
		tmp = sin(th) * (ky / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.178], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-91], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -0.17799999999999999

   1. Initial program 99.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l/ 99.7%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. clear-num 99.4%

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      3. unpow2 99.4%

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}} \]

      4. unpow2 99.4%

         \[\leadsto \frac{1}{\frac{\sqrt{\sin kx \cdot \sin kx + \color{blue}{\sin ky \cdot \sin ky}}}{\sin ky \cdot \sin th}} \]

      5. hypot-def 99.4%

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{\sin ky \cdot \sin th}} \]

   4. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot \sin th}}} \]

   5. Taylor expanded in kx around 0 3.1%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\sin th}}} \]
   6. Step-by-step derivation

      1. remove-double-div 3.1%

         \[\leadsto \color{blue}{\sin th} \]

      2. add-sqr-sqrt 1.8%

         \[\leadsto \color{blue}{\sqrt{\sin th} \cdot \sqrt{\sin th}} \]

      3. sqrt-unprod 22.2%

         \[\leadsto \color{blue}{\sqrt{\sin th \cdot \sin th}} \]

      4. pow2 22.2%

         \[\leadsto \sqrt{\color{blue}{{\sin th}^{2}}} \]

   7. Applied egg-rr 22.2%

      \[\leadsto \color{blue}{\sqrt{{\sin th}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 22.2%

         \[\leadsto \sqrt{\color{blue}{\sin th \cdot \sin th}} \]

      2. rem-sqrt-square 29.7%

         \[\leadsto \color{blue}{\left|\sin th\right|} \]

   9. Simplified 29.7%

      \[\leadsto \color{blue}{\left|\sin th\right|} \]

   if -0.17799999999999999 < (sin.f64 ky) < 4.99999999999999997e-91

   1. Initial program 88.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0 50.1%

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sin kx}} \]
   4. Step-by-step derivation

      1. associate-/l* 50.0%

         \[\leadsto \color{blue}{\frac{ky}{\frac{\sin kx}{\sin th}}} \]

      2. associate-/r/ 49.9%

         \[\leadsto \color{blue}{\frac{ky}{\sin kx} \cdot \sin th} \]

   5. Simplified 49.9%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx} \cdot \sin th} \]

   if 4.99999999999999997e-91 < (sin.f64 ky)

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0 62.8%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.178:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-91}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 46.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.178:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-91}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.178)
   (fabs (sin th))
   (if (<= (sin ky) 5e-91) (/ ky (/ (sin kx) (sin th))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.178) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 5e-91) {
		tmp = ky / (sin(kx) / sin(th));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-0.178d0)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 5d-91) then
        tmp = ky / (sin(kx) / sin(th))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.178) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 5e-91) {
		tmp = ky / (Math.sin(kx) / Math.sin(th));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.178:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 5e-91:
		tmp = ky / (math.sin(kx) / math.sin(th))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.178)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 5e-91)
		tmp = Float64(ky / Float64(sin(kx) / sin(th)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.178)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 5e-91)
		tmp = ky / (sin(kx) / sin(th));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.178], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-91], N[(ky / N[(N[Sin[kx], $MachinePrecision] / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -0.17799999999999999

   1. Initial program 99.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l/ 99.7%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. clear-num 99.4%

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      3. unpow2 99.4%

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}} \]

      4. unpow2 99.4%

         \[\leadsto \frac{1}{\frac{\sqrt{\sin kx \cdot \sin kx + \color{blue}{\sin ky \cdot \sin ky}}}{\sin ky \cdot \sin th}} \]

      5. hypot-def 99.4%

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{\sin ky \cdot \sin th}} \]

   4. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot \sin th}}} \]

   5. Taylor expanded in kx around 0 3.1%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\sin th}}} \]
   6. Step-by-step derivation

      1. remove-double-div 3.1%

         \[\leadsto \color{blue}{\sin th} \]

      2. add-sqr-sqrt 1.8%

         \[\leadsto \color{blue}{\sqrt{\sin th} \cdot \sqrt{\sin th}} \]

      3. sqrt-unprod 22.2%

         \[\leadsto \color{blue}{\sqrt{\sin th \cdot \sin th}} \]

      4. pow2 22.2%

         \[\leadsto \sqrt{\color{blue}{{\sin th}^{2}}} \]

   7. Applied egg-rr 22.2%

      \[\leadsto \color{blue}{\sqrt{{\sin th}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 22.2%

         \[\leadsto \sqrt{\color{blue}{\sin th \cdot \sin th}} \]

      2. rem-sqrt-square 29.7%

         \[\leadsto \color{blue}{\left|\sin th\right|} \]

   9. Simplified 29.7%

      \[\leadsto \color{blue}{\left|\sin th\right|} \]

   if -0.17799999999999999 < (sin.f64 ky) < 4.99999999999999997e-91

   1. Initial program 88.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0 50.1%

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sin kx}} \]
   4. Step-by-step derivation

      1. associate-/l* 50.0%

         \[\leadsto \color{blue}{\frac{ky}{\frac{\sin kx}{\sin th}}} \]

   5. Simplified 50.0%

      \[\leadsto \color{blue}{\frac{ky}{\frac{\sin kx}{\sin th}}} \]

   if 4.99999999999999997e-91 < (sin.f64 ky)

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0 62.8%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.178:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-91}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 46.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.178:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-91}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.178)
   (fabs (sin th))
   (if (<= (sin ky) 5e-91) (/ (sin th) (/ (sin kx) ky)) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.178) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 5e-91) {
		tmp = sin(th) / (sin(kx) / ky);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-0.178d0)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 5d-91) then
        tmp = sin(th) / (sin(kx) / ky)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.178) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 5e-91) {
		tmp = Math.sin(th) / (Math.sin(kx) / ky);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.178:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 5e-91:
		tmp = math.sin(th) / (math.sin(kx) / ky)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.178)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 5e-91)
		tmp = Float64(sin(th) / Float64(sin(kx) / ky));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.178)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 5e-91)
		tmp = sin(th) / (sin(kx) / ky);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.178], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-91], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -0.17799999999999999

   1. Initial program 99.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l/ 99.7%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. clear-num 99.4%

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      3. unpow2 99.4%

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}} \]

      4. unpow2 99.4%

         \[\leadsto \frac{1}{\frac{\sqrt{\sin kx \cdot \sin kx + \color{blue}{\sin ky \cdot \sin ky}}}{\sin ky \cdot \sin th}} \]

      5. hypot-def 99.4%

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{\sin ky \cdot \sin th}} \]

   4. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot \sin th}}} \]

   5. Taylor expanded in kx around 0 3.1%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\sin th}}} \]
   6. Step-by-step derivation

      1. remove-double-div 3.1%

         \[\leadsto \color{blue}{\sin th} \]

      2. add-sqr-sqrt 1.8%

         \[\leadsto \color{blue}{\sqrt{\sin th} \cdot \sqrt{\sin th}} \]

      3. sqrt-unprod 22.2%

         \[\leadsto \color{blue}{\sqrt{\sin th \cdot \sin th}} \]

      4. pow2 22.2%

         \[\leadsto \sqrt{\color{blue}{{\sin th}^{2}}} \]

   7. Applied egg-rr 22.2%

      \[\leadsto \color{blue}{\sqrt{{\sin th}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 22.2%

         \[\leadsto \sqrt{\color{blue}{\sin th \cdot \sin th}} \]

      2. rem-sqrt-square 29.7%

         \[\leadsto \color{blue}{\left|\sin th\right|} \]

   9. Simplified 29.7%

      \[\leadsto \color{blue}{\left|\sin th\right|} \]

   if -0.17799999999999999 < (sin.f64 ky) < 4.99999999999999997e-91

   1. Initial program 88.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 88.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \cdot \sin th \]

      2. associate-*l/ 88.2%

         \[\leadsto \color{blue}{\frac{1 \cdot \sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      3. *-un-lft-identity 88.2%

         \[\leadsto \frac{\color{blue}{\sin th}}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}} \]

      4. unpow2 88.2%

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}}{\sin ky}} \]

      5. unpow2 88.2%

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\sin kx \cdot \sin kx + \color{blue}{\sin ky \cdot \sin ky}}}{\sin ky}} \]

      6. hypot-def 99.7%

         \[\leadsto \frac{\sin th}{\frac{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{\sin ky}} \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}} \]

   5. Taylor expanded in ky around 0 50.0%

      \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sin kx}{ky}}} \]

   if 4.99999999999999997e-91 < (sin.f64 ky)

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0 62.8%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.178:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-91}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 45.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;\frac{1}{\sqrt{{\sin th}^{-2}}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-91}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.02)
   (/ 1.0 (sqrt (pow (sin th) -2.0)))
   (if (<= (sin ky) 5e-91) (/ (sin th) (/ (sin kx) ky)) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.02) {
		tmp = 1.0 / sqrt(pow(sin(th), -2.0));
	} else if (sin(ky) <= 5e-91) {
		tmp = sin(th) / (sin(kx) / ky);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-0.02d0)) then
        tmp = 1.0d0 / sqrt((sin(th) ** (-2.0d0)))
    else if (sin(ky) <= 5d-91) then
        tmp = sin(th) / (sin(kx) / ky)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.02) {
		tmp = 1.0 / Math.sqrt(Math.pow(Math.sin(th), -2.0));
	} else if (Math.sin(ky) <= 5e-91) {
		tmp = Math.sin(th) / (Math.sin(kx) / ky);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.02:
		tmp = 1.0 / math.sqrt(math.pow(math.sin(th), -2.0))
	elif math.sin(ky) <= 5e-91:
		tmp = math.sin(th) / (math.sin(kx) / ky)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.02)
		tmp = Float64(1.0 / sqrt((sin(th) ^ -2.0)));
	elseif (sin(ky) <= 5e-91)
		tmp = Float64(sin(th) / Float64(sin(kx) / ky));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.02)
		tmp = 1.0 / sqrt((sin(th) ^ -2.0));
	elseif (sin(ky) <= 5e-91)
		tmp = sin(th) / (sin(kx) / ky);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.02], N[(1.0 / N[Sqrt[N[Power[N[Sin[th], $MachinePrecision], -2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-91], N[(N[Sin[th], $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -0.0200000000000000004

   1. Initial program 99.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l/ 99.7%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. clear-num 99.4%

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      3. unpow2 99.4%

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}} \]

      4. unpow2 99.4%

         \[\leadsto \frac{1}{\frac{\sqrt{\sin kx \cdot \sin kx + \color{blue}{\sin ky \cdot \sin ky}}}{\sin ky \cdot \sin th}} \]

      5. hypot-def 99.4%

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{\sin ky \cdot \sin th}} \]

   4. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot \sin th}}} \]

   5. Taylor expanded in kx around 0 3.1%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\sin th}}} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 1.8%

         \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{1}{\sin th}} \cdot \sqrt{\frac{1}{\sin th}}}} \]

      2. sqrt-unprod 22.7%

         \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{1}{\sin th} \cdot \frac{1}{\sin th}}}} \]

      3. inv-pow 22.7%

         \[\leadsto \frac{1}{\sqrt{\color{blue}{{\sin th}^{-1}} \cdot \frac{1}{\sin th}}} \]

      4. inv-pow 22.7%

         \[\leadsto \frac{1}{\sqrt{{\sin th}^{-1} \cdot \color{blue}{{\sin th}^{-1}}}} \]

      5. pow-prod-up 22.7%

         \[\leadsto \frac{1}{\sqrt{\color{blue}{{\sin th}^{\left(-1 + -1\right)}}}} \]

      6. metadata-eval 22.7%

         \[\leadsto \frac{1}{\sqrt{{\sin th}^{\color{blue}{-2}}}} \]

   7. Applied egg-rr 22.7%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{{\sin th}^{-2}}}} \]

   if -0.0200000000000000004 < (sin.f64 ky) < 4.99999999999999997e-91

   1. Initial program 87.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 87.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \cdot \sin th \]

      2. associate-*l/ 87.7%

         \[\leadsto \color{blue}{\frac{1 \cdot \sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      3. *-un-lft-identity 87.7%

         \[\leadsto \frac{\color{blue}{\sin th}}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}} \]

      4. unpow2 87.7%

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}}{\sin ky}} \]

      5. unpow2 87.7%

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\sin kx \cdot \sin kx + \color{blue}{\sin ky \cdot \sin ky}}}{\sin ky}} \]

      6. hypot-def 99.7%

         \[\leadsto \frac{\sin th}{\frac{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{\sin ky}} \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}} \]

   5. Taylor expanded in ky around 0 52.2%

      \[\leadsto \frac{\sin th}{\color{blue}{\frac{\sin kx}{ky}}} \]

   if 4.99999999999999997e-91 < (sin.f64 ky)

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0 62.8%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;\frac{1}{\sqrt{{\sin th}^{-2}}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-91}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 30.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 2.4 \cdot 10^{-162}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;kx \leq 1.2 \cdot 10^{-55}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left|\frac{\sin ky}{\sin kx}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 2.4e-162)
   (sin th)
   (if (<= kx 1.2e-55)
     (fabs (sin th))
     (* (sin th) (fabs (/ (sin ky) (sin kx)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 2.4e-162) {
		tmp = sin(th);
	} else if (kx <= 1.2e-55) {
		tmp = fabs(sin(th));
	} else {
		tmp = sin(th) * fabs((sin(ky) / sin(kx)));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (kx <= 2.4d-162) then
        tmp = sin(th)
    else if (kx <= 1.2d-55) then
        tmp = abs(sin(th))
    else
        tmp = sin(th) * abs((sin(ky) / sin(kx)))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 2.4e-162) {
		tmp = Math.sin(th);
	} else if (kx <= 1.2e-55) {
		tmp = Math.abs(Math.sin(th));
	} else {
		tmp = Math.sin(th) * Math.abs((Math.sin(ky) / Math.sin(kx)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 2.4e-162:
		tmp = math.sin(th)
	elif kx <= 1.2e-55:
		tmp = math.fabs(math.sin(th))
	else:
		tmp = math.sin(th) * math.fabs((math.sin(ky) / math.sin(kx)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 2.4e-162)
		tmp = sin(th);
	elseif (kx <= 1.2e-55)
		tmp = abs(sin(th));
	else
		tmp = Float64(sin(th) * abs(Float64(sin(ky) / sin(kx))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 2.4e-162)
		tmp = sin(th);
	elseif (kx <= 1.2e-55)
		tmp = abs(sin(th));
	else
		tmp = sin(th) * abs((sin(ky) / sin(kx)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 2.4e-162], N[Sin[th], $MachinePrecision], If[LessEqual[kx, 1.2e-55], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[Abs[N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if kx < 2.4000000000000002e-162

   1. Initial program 92.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0 31.5%

      \[\leadsto \color{blue}{\sin th} \]

   if 2.4000000000000002e-162 < kx < 1.19999999999999996e-55

   1. Initial program 99.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l/ 95.5%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. clear-num 95.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      3. unpow2 95.2%

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}} \]

      4. unpow2 95.2%

         \[\leadsto \frac{1}{\frac{\sqrt{\sin kx \cdot \sin kx + \color{blue}{\sin ky \cdot \sin ky}}}{\sin ky \cdot \sin th}} \]

      5. hypot-def 95.2%

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{\sin ky \cdot \sin th}} \]

   4. Applied egg-rr 95.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot \sin th}}} \]

   5. Taylor expanded in kx around 0 36.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\sin th}}} \]
   6. Step-by-step derivation

      1. remove-double-div 36.1%

         \[\leadsto \color{blue}{\sin th} \]

      2. add-sqr-sqrt 30.2%

         \[\leadsto \color{blue}{\sqrt{\sin th} \cdot \sqrt{\sin th}} \]

      3. sqrt-unprod 23.4%

         \[\leadsto \color{blue}{\sqrt{\sin th \cdot \sin th}} \]

      4. pow2 23.4%

         \[\leadsto \sqrt{\color{blue}{{\sin th}^{2}}} \]

   7. Applied egg-rr 23.4%

      \[\leadsto \color{blue}{\sqrt{{\sin th}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 23.4%

         \[\leadsto \sqrt{\color{blue}{\sin th \cdot \sin th}} \]

      2. rem-sqrt-square 45.3%

         \[\leadsto \color{blue}{\left|\sin th\right|} \]

   9. Simplified 45.3%

      \[\leadsto \color{blue}{\left|\sin th\right|} \]

   if 1.19999999999999996e-55 < kx

   1. Initial program 99.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0 45.6%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 25.0%

         \[\leadsto \color{blue}{\left(\sqrt{\frac{\sin ky}{\sin kx}} \cdot \sqrt{\frac{\sin ky}{\sin kx}}\right)} \cdot \sin th \]

      2. sqrt-unprod 33.7%

         \[\leadsto \color{blue}{\sqrt{\frac{\sin ky}{\sin kx} \cdot \frac{\sin ky}{\sin kx}}} \cdot \sin th \]

      3. pow2 33.7%

         \[\leadsto \sqrt{\color{blue}{{\left(\frac{\sin ky}{\sin kx}\right)}^{2}}} \cdot \sin th \]

   5. Applied egg-rr 33.7%

      \[\leadsto \color{blue}{\sqrt{{\left(\frac{\sin ky}{\sin kx}\right)}^{2}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. unpow2 33.7%

         \[\leadsto \sqrt{\color{blue}{\frac{\sin ky}{\sin kx} \cdot \frac{\sin ky}{\sin kx}}} \cdot \sin th \]

      2. rem-sqrt-square 39.1%

         \[\leadsto \color{blue}{\left|\frac{\sin ky}{\sin kx}\right|} \cdot \sin th \]

   7. Simplified 39.1%

      \[\leadsto \color{blue}{\left|\frac{\sin ky}{\sin kx}\right|} \cdot \sin th \]
3. Recombined 3 regimes into one program.

4. Final simplification 34.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 2.4 \cdot 10^{-162}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;kx \leq 1.2 \cdot 10^{-55}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left|\frac{\sin ky}{\sin kx}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 30.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 4.7 \cdot 10^{-163}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;kx \leq 5 \cdot 10^{-57}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\left|\frac{\sin kx}{\sin ky}\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 4.7e-163)
   (sin th)
   (if (<= kx 5e-57)
     (fabs (sin th))
     (/ (sin th) (fabs (/ (sin kx) (sin ky)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 4.7e-163) {
		tmp = sin(th);
	} else if (kx <= 5e-57) {
		tmp = fabs(sin(th));
	} else {
		tmp = sin(th) / fabs((sin(kx) / sin(ky)));
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (kx <= 4.7d-163) then
        tmp = sin(th)
    else if (kx <= 5d-57) then
        tmp = abs(sin(th))
    else
        tmp = sin(th) / abs((sin(kx) / sin(ky)))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 4.7e-163) {
		tmp = Math.sin(th);
	} else if (kx <= 5e-57) {
		tmp = Math.abs(Math.sin(th));
	} else {
		tmp = Math.sin(th) / Math.abs((Math.sin(kx) / Math.sin(ky)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 4.7e-163:
		tmp = math.sin(th)
	elif kx <= 5e-57:
		tmp = math.fabs(math.sin(th))
	else:
		tmp = math.sin(th) / math.fabs((math.sin(kx) / math.sin(ky)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 4.7e-163)
		tmp = sin(th);
	elseif (kx <= 5e-57)
		tmp = abs(sin(th));
	else
		tmp = Float64(sin(th) / abs(Float64(sin(kx) / sin(ky))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 4.7e-163)
		tmp = sin(th);
	elseif (kx <= 5e-57)
		tmp = abs(sin(th));
	else
		tmp = sin(th) / abs((sin(kx) / sin(ky)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 4.7e-163], N[Sin[th], $MachinePrecision], If[LessEqual[kx, 5e-57], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[Abs[N[(N[Sin[kx], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if kx < 4.7e-163

   1. Initial program 92.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0 31.5%

      \[\leadsto \color{blue}{\sin th} \]

   if 4.7e-163 < kx < 5.0000000000000002e-57

   1. Initial program 99.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l/ 95.5%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. clear-num 95.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      3. unpow2 95.2%

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}} \]

      4. unpow2 95.2%

         \[\leadsto \frac{1}{\frac{\sqrt{\sin kx \cdot \sin kx + \color{blue}{\sin ky \cdot \sin ky}}}{\sin ky \cdot \sin th}} \]

      5. hypot-def 95.2%

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{\sin ky \cdot \sin th}} \]

   4. Applied egg-rr 95.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot \sin th}}} \]

   5. Taylor expanded in kx around 0 36.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\sin th}}} \]
   6. Step-by-step derivation

      1. remove-double-div 36.1%

         \[\leadsto \color{blue}{\sin th} \]

      2. add-sqr-sqrt 30.2%

         \[\leadsto \color{blue}{\sqrt{\sin th} \cdot \sqrt{\sin th}} \]

      3. sqrt-unprod 23.4%

         \[\leadsto \color{blue}{\sqrt{\sin th \cdot \sin th}} \]

      4. pow2 23.4%

         \[\leadsto \sqrt{\color{blue}{{\sin th}^{2}}} \]

   7. Applied egg-rr 23.4%

      \[\leadsto \color{blue}{\sqrt{{\sin th}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 23.4%

         \[\leadsto \sqrt{\color{blue}{\sin th \cdot \sin th}} \]

      2. rem-sqrt-square 45.3%

         \[\leadsto \color{blue}{\left|\sin th\right|} \]

   9. Simplified 45.3%

      \[\leadsto \color{blue}{\left|\sin th\right|} \]

   if 5.0000000000000002e-57 < kx

   1. Initial program 99.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 99.4%

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \cdot \sin th \]

      2. associate-*l/ 99.6%

         \[\leadsto \color{blue}{\frac{1 \cdot \sin th}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}}} \]

      3. *-un-lft-identity 99.6%

         \[\leadsto \frac{\color{blue}{\sin th}}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky}} \]

      4. unpow2 99.6%

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}}{\sin ky}} \]

      5. unpow2 99.6%

         \[\leadsto \frac{\sin th}{\frac{\sqrt{\sin kx \cdot \sin kx + \color{blue}{\sin ky \cdot \sin ky}}}{\sin ky}} \]

      6. hypot-def 99.6%

         \[\leadsto \frac{\sin th}{\frac{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{\sin ky}} \]

   4. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}} \]

   5. Taylor expanded in ky around 0 45.7%

      \[\leadsto \frac{\sin th}{\frac{\color{blue}{\sin kx}}{\sin ky}} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 25.1%

         \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{\frac{\sin kx}{\sin ky}} \cdot \sqrt{\frac{\sin kx}{\sin ky}}}} \]

      2. sqrt-unprod 32.8%

         \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{\frac{\sin kx}{\sin ky} \cdot \frac{\sin kx}{\sin ky}}}} \]

      3. pow2 32.8%

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{{\left(\frac{\sin kx}{\sin ky}\right)}^{2}}}} \]

   7. Applied egg-rr 32.8%

      \[\leadsto \frac{\sin th}{\color{blue}{\sqrt{{\left(\frac{\sin kx}{\sin ky}\right)}^{2}}}} \]
   8. Step-by-step derivation

      1. unpow2 32.8%

         \[\leadsto \frac{\sin th}{\sqrt{\color{blue}{\frac{\sin kx}{\sin ky} \cdot \frac{\sin kx}{\sin ky}}}} \]

      2. rem-sqrt-square 39.1%

         \[\leadsto \frac{\sin th}{\color{blue}{\left|\frac{\sin kx}{\sin ky}\right|}} \]

   9. Simplified 39.1%

      \[\leadsto \frac{\sin th}{\color{blue}{\left|\frac{\sin kx}{\sin ky}\right|}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 34.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 4.7 \cdot 10^{-163}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;kx \leq 5 \cdot 10^{-57}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\left|\frac{\sin kx}{\sin ky}\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 40.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-151}:\\ \;\;\;\;\frac{ky}{\frac{kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.02)
   (fabs (sin th))
   (if (<= (sin ky) 5e-151) (/ ky (/ kx (sin th))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.02) {
		tmp = fabs(sin(th));
	} else if (sin(ky) <= 5e-151) {
		tmp = ky / (kx / sin(th));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= (-0.02d0)) then
        tmp = abs(sin(th))
    else if (sin(ky) <= 5d-151) then
        tmp = ky / (kx / sin(th))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.02) {
		tmp = Math.abs(Math.sin(th));
	} else if (Math.sin(ky) <= 5e-151) {
		tmp = ky / (kx / Math.sin(th));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.02:
		tmp = math.fabs(math.sin(th))
	elif math.sin(ky) <= 5e-151:
		tmp = ky / (kx / math.sin(th))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.02)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 5e-151)
		tmp = Float64(ky / Float64(kx / sin(th)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.02)
		tmp = abs(sin(th));
	elseif (sin(ky) <= 5e-151)
		tmp = ky / (kx / sin(th));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.02], N[Abs[N[Sin[th], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-151], N[(ky / N[(kx / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -0.0200000000000000004

   1. Initial program 99.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l/ 99.7%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. clear-num 99.4%

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

      3. unpow2 99.4%

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}} \]

      4. unpow2 99.4%

         \[\leadsto \frac{1}{\frac{\sqrt{\sin kx \cdot \sin kx + \color{blue}{\sin ky \cdot \sin ky}}}{\sin ky \cdot \sin th}} \]

      5. hypot-def 99.4%

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{\sin ky \cdot \sin th}} \]

   4. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot \sin th}}} \]

   5. Taylor expanded in kx around 0 3.1%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\sin th}}} \]
   6. Step-by-step derivation

      1. remove-double-div 3.1%

         \[\leadsto \color{blue}{\sin th} \]

      2. add-sqr-sqrt 1.8%

         \[\leadsto \color{blue}{\sqrt{\sin th} \cdot \sqrt{\sin th}} \]

      3. sqrt-unprod 22.6%

         \[\leadsto \color{blue}{\sqrt{\sin th \cdot \sin th}} \]

      4. pow2 22.6%

         \[\leadsto \sqrt{\color{blue}{{\sin th}^{2}}} \]

   7. Applied egg-rr 22.6%

      \[\leadsto \color{blue}{\sqrt{{\sin th}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 22.6%

         \[\leadsto \sqrt{\color{blue}{\sin th \cdot \sin th}} \]

      2. rem-sqrt-square 31.0%

         \[\leadsto \color{blue}{\left|\sin th\right|} \]

   9. Simplified 31.0%

      \[\leadsto \color{blue}{\left|\sin th\right|} \]

   if -0.0200000000000000004 < (sin.f64 ky) < 5.00000000000000003e-151

   1. Initial program 85.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0 54.9%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   4. Taylor expanded in kx around 0 38.6%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{kx}} \]

   5. Taylor expanded in ky around 0 38.6%

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{kx}} \]
   6. Step-by-step derivation

      1. associate-/l* 38.4%

         \[\leadsto \color{blue}{\frac{ky}{\frac{kx}{\sin th}}} \]

   7. Simplified 38.4%

      \[\leadsto \color{blue}{\frac{ky}{\frac{kx}{\sin th}}} \]

   if 5.00000000000000003e-151 < (sin.f64 ky)

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0 58.7%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-151}:\\ \;\;\;\;\frac{ky}{\frac{kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 32.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq 5 \cdot 10^{-151}:\\ \;\;\;\;\frac{ky}{\frac{kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) 5e-151) (/ ky (/ kx (sin th))) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= 5e-151) {
		tmp = ky / (kx / sin(th));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (sin(ky) <= 5d-151) then
        tmp = ky / (kx / sin(th))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= 5e-151) {
		tmp = ky / (kx / Math.sin(th));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= 5e-151:
		tmp = ky / (kx / math.sin(th))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= 5e-151)
		tmp = Float64(ky / Float64(kx / sin(th)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= 5e-151)
		tmp = ky / (kx / sin(th));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], 5e-151], N[(ky / N[(kx / N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < 5.00000000000000003e-151

   1. Initial program 91.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0 33.3%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   4. Taylor expanded in kx around 0 23.5%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{kx}} \]

   5. Taylor expanded in ky around 0 22.9%

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{kx}} \]
   6. Step-by-step derivation

      1. associate-/l* 22.9%

         \[\leadsto \color{blue}{\frac{ky}{\frac{kx}{\sin th}}} \]

   7. Simplified 22.9%

      \[\leadsto \color{blue}{\frac{ky}{\frac{kx}{\sin th}}} \]

   if 5.00000000000000003e-151 < (sin.f64 ky)

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0 58.7%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin ky \leq 5 \cdot 10^{-151}:\\ \;\;\;\;\frac{ky}{\frac{kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 22.3% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 3 \cdot 10^{-152}:\\ \;\;\;\;\frac{0}{\sin kx \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 3e-152) (/ 0.0 (* (sin kx) 2.0)) (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 3e-152) {
		tmp = 0.0 / (sin(kx) * 2.0);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (ky <= 3d-152) then
        tmp = 0.0d0 / (sin(kx) * 2.0d0)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 3e-152) {
		tmp = 0.0 / (Math.sin(kx) * 2.0);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 3e-152:
		tmp = 0.0 / (math.sin(kx) * 2.0)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 3e-152)
		tmp = Float64(0.0 / Float64(sin(kx) * 2.0));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 3e-152)
		tmp = 0.0 / (sin(kx) * 2.0);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 3e-152], N[(0.0 / N[(N[Sin[kx], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 3e-152

   1. Initial program 91.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0 34.2%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. associate-*l/ 34.3%

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sin kx}} \]

      2. sin-mult 17.7%

         \[\leadsto \frac{\color{blue}{\frac{\cos \left(ky - th\right) - \cos \left(ky + th\right)}{2}}}{\sin kx} \]

      3. associate-/l/ 17.7%

         \[\leadsto \color{blue}{\frac{\cos \left(ky - th\right) - \cos \left(ky + th\right)}{\sin kx \cdot 2}} \]

      4. +-commutative 17.7%

         \[\leadsto \frac{\cos \left(ky - th\right) - \cos \color{blue}{\left(th + ky\right)}}{\sin kx \cdot 2} \]

   5. Applied egg-rr 17.7%

      \[\leadsto \color{blue}{\frac{\cos \left(ky - th\right) - \cos \left(th + ky\right)}{\sin kx \cdot 2}} \]

   6. Taylor expanded in ky around 0 17.6%

      \[\leadsto \frac{\color{blue}{\cos \left(-th\right) - \cos th}}{\sin kx \cdot 2} \]
   7. Step-by-step derivation

      1. cos-neg 17.6%

         \[\leadsto \frac{\color{blue}{\cos th} - \cos th}{\sin kx \cdot 2} \]

      2. +-inverses 17.6%

         \[\leadsto \frac{\color{blue}{0}}{\sin kx \cdot 2} \]

   8. Simplified 17.6%

      \[\leadsto \frac{\color{blue}{0}}{\sin kx \cdot 2} \]

   if 3e-152 < ky

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0 38.2%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 25.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 3 \cdot 10^{-152}:\\ \;\;\;\;\frac{0}{\sin kx \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 23.2% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \sin th \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 (sin th))

C

double code(double kx, double ky, double th) {
	return sin(th);
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = sin(th)
end function

Java

public static double code(double kx, double ky, double th) {
	return Math.sin(th);
}

Python

def code(kx, ky, th):
	return math.sin(th)

Julia

function code(kx, ky, th)
	return sin(th)
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = sin(th);
end

Wolfram

code[kx_, ky_, th_] := N[Sin[th], $MachinePrecision]

Derivation

1. Initial program 94.7%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing

3. Taylor expanded in kx around 0 25.9%

   \[\leadsto \color{blue}{\sin th} \]

4. Final simplification 25.9%

   \[\leadsto \sin th \]
5. Add Preprocessing

Alternative 20: 14.1% accurate, 78.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{th \cdot 0.16666666666666666 + \frac{1}{th}} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (/ 1.0 (+ (* th 0.16666666666666666) (/ 1.0 th))))

C

double code(double kx, double ky, double th) {
	return 1.0 / ((th * 0.16666666666666666) + (1.0 / th));
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = 1.0d0 / ((th * 0.16666666666666666d0) + (1.0d0 / th))
end function

Java

public static double code(double kx, double ky, double th) {
	return 1.0 / ((th * 0.16666666666666666) + (1.0 / th));
}

Python

def code(kx, ky, th):
	return 1.0 / ((th * 0.16666666666666666) + (1.0 / th))

Julia

function code(kx, ky, th)
	return Float64(1.0 / Float64(Float64(th * 0.16666666666666666) + Float64(1.0 / th)))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = 1.0 / ((th * 0.16666666666666666) + (1.0 / th));
end

Wolfram

code[kx_, ky_, th_] := N[(1.0 / N[(N[(th * 0.16666666666666666), $MachinePrecision] + N[(1.0 / th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.7%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l/ 93.6%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

   2. clear-num 93.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

   3. unpow2 93.2%

      \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}} \]

   4. unpow2 93.2%

      \[\leadsto \frac{1}{\frac{\sqrt{\sin kx \cdot \sin kx + \color{blue}{\sin ky \cdot \sin ky}}}{\sin ky \cdot \sin th}} \]

   5. hypot-def 97.1%

      \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{\sin ky \cdot \sin th}} \]

4. Applied egg-rr 97.1%

   \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot \sin th}}} \]

5. Taylor expanded in kx around 0 25.9%

   \[\leadsto \frac{1}{\color{blue}{\frac{1}{\sin th}}} \]

6. Taylor expanded in th around 0 15.6%

   \[\leadsto \frac{1}{\color{blue}{0.16666666666666666 \cdot th + \frac{1}{th}}} \]

7. Final simplification 15.6%

   \[\leadsto \frac{1}{th \cdot 0.16666666666666666 + \frac{1}{th}} \]
8. Add Preprocessing

Alternative 21: 13.4% accurate, 709.0× speedup

Math

\[\begin{array}{l} \\ th \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 th)

C

double code(double kx, double ky, double th) {
	return th;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = th
end function

Java

public static double code(double kx, double ky, double th) {
	return th;
}

Python

def code(kx, ky, th):
	return th

Julia

function code(kx, ky, th)
	return th
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = th;
end

Wolfram

code[kx_, ky_, th_] := th

Derivation

1. Initial program 94.7%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l/ 93.6%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

   2. clear-num 93.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}}} \]

   3. unpow2 93.2%

      \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}}{\sin ky \cdot \sin th}} \]

   4. unpow2 93.2%

      \[\leadsto \frac{1}{\frac{\sqrt{\sin kx \cdot \sin kx + \color{blue}{\sin ky \cdot \sin ky}}}{\sin ky \cdot \sin th}} \]

   5. hypot-def 97.1%

      \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}{\sin ky \cdot \sin th}} \]

4. Applied egg-rr 97.1%

   \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky \cdot \sin th}}} \]

5. Taylor expanded in kx around 0 25.9%

   \[\leadsto \frac{1}{\color{blue}{\frac{1}{\sin th}}} \]

6. Taylor expanded in th around 0 14.9%

   \[\leadsto \color{blue}{th} \]

7. Final simplification 14.9%

   \[\leadsto th \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024014 
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  :precision binary64
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) (sin th)))